Self-Dual Gravity Revisited

TL;DR

This paper revisits the harmonic space approach to self-dual Einstein equations, providing a streamlined proof and a practical method for constructing self-dual metrics from an unconstrained prepotential, exemplified by the Taub-NUT metric.

Contribution

It offers a simplified proof that all self-dual Einstein fields can be described by an unconstrained prepotential in harmonic space and provides a practical construction method.

Findings

Streamlined proof of the harmonic space description for self-dual Einstein equations.

A recipe for constructing self-dual metrics from a chosen prepotential.

Explicit example of deriving the Taub-NUT metric from a simple prepotential.

Abstract

Reconsidering the harmonic space description of the self-dual Einstein equations, we streamline the proof that all self-dual pure gravitational fields allow a local description in terms of an unconstrained analytic prepotential in harmonic space. Our formulation yields a simple recipe for constructing self-dual metrics starting from any explicit choice of such prepotential; and we illustrate the procedure by producing a metric related to the Taub-NUT solution from the simplest monomial choice of prepotential.